2020/08/28 by Jérôme Lemoine, Lemoine, Jerome, Irene Marín-Gayte +3
Engineering · Mathematics · #35Q30 #93E24 #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods for differential equations #Numerical methods in inverse problems #Optimization and Control (math.OC) #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2008.12656
openalex publication_date 2020/08/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The null distributed controllability of the semilinear heat equation\nyt-\Δ y + g(y)=f ,1\ω, assuming that g satisfies the growth\ncondition g(s)/( vert s vert \log3/2(1+ vert s vert))\→ 0 as\n vert s vert \→ \∞ and that g^\′\∈\nL^\∞loc(\ℝ) has been obtained by Fern 'andez-Cara and Zuazua in\n2000. The proof based on a fixed point argument makes use of precise estimates\nof the observability constant for a linearized heat equation. It does not\nprovide however an explicit construction of a null control. Assuming that\ng^\′\∈ Ws,\∞(\ℝ) for one s\∈ (0,1], we construct an\nexplicit sequence converging strongly to a null control for the solution of the\nsemilinear equation. The method, based on a least-squares approach, generalizes\nNewton type methods and guarantees the convergence whatever be the initial\nelement of the sequence. In particular, after a finite number of iterations,\nthe convergence is super linear with a rate equal to 1+s. Numerical\nexperiments in the one dimensional setting support our analysis.\n